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Parikh vectors
1NIM_1 2OTP_1 3WVF_1 Letter Amino acid
0 6 27 N Asparagine
0 9 6 H Histidine
0 14 30 V Valine
1 4 28 F Phenylalanine
1 17 34 P Proline
0 9 14 R Arginine
2 6 1 C Cysteine
1 12 20 E Glutamic acid
0 18 57 L Leucine
3 5 26 K Lycine
0 0 22 M Methionine
1 7 39 T Threonine
2 5 24 D Aspartic acid
2 9 43 Q Glutamine
2 25 29 S Serine
0 7 11 W Tryptophan
0 11 25 Y Tyrosine
0 10 41 A Alanine
1 15 39 G Glycine
1 7 35 I Isoleucine 

1NIM_1|Chain A|PAK PILIN, TRANS|Pseudomonas aeruginosa (287)
>2OTP_1|Chains A, B|Leukocyte immunoglobulin-like receptor subfamily A member 2|Homo sapiens (9606)
>3WVF_1|Chains A, B|Membrane protein insertase YidC|Escherichia coli (364106)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
1NIM , Knot 14 18 0.75 24 17 16 
XKCTSDQDEQFIPKGCSK
2OTP , Knot 91 196 0.81 38 136 184 
GHLPKPTLWAEPGSVIIQGSPVTLRCQGSLQAEEYHLYRENKSASWVRRIQEPGKNGQFPIPSITWEHAGRYHCQYYSHNHSSEYSDPLELVVTGAYSKPTLSALPSPVVTLGGNVTLQCVSQVAFDGFILCKEGEDEHPQCLNSHSHARGWSWAIFSVGPVSPSRRWSYRCYAYDSNSPYVWSLPSDLLELLVPG
3WVF , Knot 224 551 0.85 40 263 520 
MDSQRNLLVIALLFVSFMIWQAWEQDKNPQPQAQQTTQTTTTAAGSAADQGVPASGQGKLISVKTDVLDLTINTRGGDVEQALLPAYPKELNSTQPFQLLETSPQFIYQAQSGLTGRDGPDNPANGPRPLYNVEKDAYVLAEGQNELQVPMTYTDAAGNTFTKTFVLKRGDYAVNVNYNVQNAGEKPLEISTFGQLKQSITLPPHLDTGSSNFALHTFRGAAYSTPDEKYEKYKFDTIADNENLNISSKGGWVAMLQQYFATAWIPHNDGTNNFYTANLGNGIAAIGYKSQPVLVQPGQTGAMNSTLWVGPEIQDKMAAVAPHLDLTVDYGWLWFISQPLFKLLKWIHSFVGNWGFSIIIITFIVRGIMYPLTKAQYTSMAKMRMLQPKIQAMRERLGDDKQRISQEMMALYKAEKVNPLGGCFPLLIQMPIFLALYYMLMGSVELRQAPFALWIHDLSAQDPYYILPILMGVTMFFIQKMSPTTVTDPMQQKIMTFMPVIFTVFFLWFPSGLVLYYIVSNLVTIIQQQLIYRGLEKRGLLESSGENLYFQ

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(1NIM_1)}(2) \setminus P_{f(2OTP_1)}(2)|=10\), \(|P_{f(2OTP_1)}(2) \setminus P_{f(1NIM_1)}(2)|=129\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:000000000011101000
Pair \(Z_2\) Length of longest common subsequence
1NIM_1,2OTP_1 139 2
1NIM_1,3WVF_1 256 3
2OTP_1,3WVF_1 203 3

Newick tree

 
[
	3WVF_1:12.20,
	[
		1NIM_1:69.5,2OTP_1:69.5
	]:57.70
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{214 }{\log_{20} 214}-\frac{18}{\log_{20}18})=68.5\)
Status Protein1 Protein2 d d1/2
Query variables 1NIM_1 2OTP_1 86 47
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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